Arg max

Does everybody except me know the notation “arg max?”  I learned it in an applied math talk today.  I had never seen this before.  It means “the value at which the maximum is achieved” — e.g.

arg max (-x^2 + x - 1) = 1/2.



10 thoughts on “Arg max

  1. Scott McKuen says:

    It’s more of a CS idiom, I think. Very common in machine learning applications, but I’ve never seen it in a straight math lecture.

  2. Anon. IV says:

    Seen it, yes, but … what’s argmax(x^4-x^2+1)?

  3. Anon says:

    Presumably \emptyset .

  4. Anon. IV says:

    I meant something like argmax(x^2-x^4) [or argmin(x^4-x^2+1)], for which the answer would still need to be a set of cardinality other than 1, albeit for a different reason: as it happens x^4-x^2+1 illustrates another difficulty with this notation (and could just have easily been asked about x^2, or indeed just x).

  5. Anon says:

    This is used in the dynamic programming literature; probably also in the optimization literature more generally.

  6. I think I saw that once in a number theory paper, though I don’t remember when…

  7. I just learned it yesterday too. At our departmental coffee hour, someone used it to describe the “Chebychev center” of a set, the better to discuss the collaboration graph of the department.

  8. Michael Lugo says:

    I associate it with CS. There are definitely times where I’ve wanted to use it in straight math contexts, but I tend to refrain from doing so.

  9. Lior Silberman says:

    I thought it was a well-known notation. Here’s one useful instance: take a bounded set A in Euclidean space (more generally, a probably measure of compact support), and let $d(z)$ be the average over $A$ of the squared distance to $z$: $d_A(z) = \int_A dx \norm{z-x}^2$). Then the center of mass of $A$ is precisely $\argmin(d_A)$.

  10. Florian says:

    I didn’t know it, but I would have learnt it today. Did we see the same talk?? (Spielman just started his special lecture series at U of T.)

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